Rational points on the superelliptic Erd?s Selfridge curve of fifth degree

?1. Introduction, By a remarkable result of Erdos and Selfridge [3] in 1975. the diophantine equationwith integers k?2 and m?2, has only the trivial solutions. x= ?j(j = i, ?, m), y= 0. This put an end to the old question whether the product of consecutive positive integers could ever be a perfect power; for a brief account of its history see [7].